HIGH ENERGY ASTROPHYSICS - Lecture 3 PD Frank Rieger ITA & MPIK Heidelberg 1 RADIATION TRANSFER (Basics) 1 Overview • Knowledge in HEA essentially depends on proper understanding of radiation received from distant objects. • This lecture introduces/recaptures some of the basic radiation properties, definitions etc. • It closes with black body radiation and discusses an application in the context of AGN. 2 2 Flux Density F Energy flux density F := Energy dE passing though area dA in time interval dt dE F := (1) dA dt Units [F ] = erg s−1 cm−2. dA Note: F depends on orientation of dA and can also depend on frequency ν, i.e. Fν := dE=[dA dt dν]. 3 3 Inverse-Square Law for Flux Density Consider: Flux from an isotropic radiation source = source emitting equal amounts of energy in all directions. Energy conservations (dE = dE1) implies that flux through two spherical surfaces is identical: 2 2 4πr F (r) = 4πr1F (r1) so that F (r )r2 const L F (r) = 1 1 = =: r2 r2 4πr2 where L is called luminosity. �� � Note: Spherically symmetric stars are isotropic emitters, other objects like AGN are generally not (apparent isotropic luminosity). 4 4 Specific Intensity Iν • Ray Optics Approximation: Energy carried along individual rays (straight lines). • Problem: Rays are infinitely thin, single ray carries essentially no energy. ) Consider set of rays: construct area dA normal to a given ray and look at all rays passing through area element within solid angle dΩ of the given ray. d� dA normal 5 • Specific Intensity, Iν, in frequency band ν; :::; ν + dν is defined via dE =: Iν dAdtdΩdν −1 −2 −1 −1 • Units [Iν] =erg s cm ster Hz . • Note: 1. Iν(ν; Ω) depends on location, direction and frequency. For isotropic radiation field, Iν =constant for all directions (independent of angle). R −1 −2 −1 2. Total integrated intensity I := Iνdν [erg s cm ster ]. 3. Specific Intensity is sometimes called spectral intensity or spectral bright- ness, or loosely just brightness. 6 5 Net Flux Fν and Specific Intensity Iν For area element dA at some arbitrary orientation ~n with respect to ray. Contribution dFν to flux in direction ~n from flux in direction of dΩ: dFν := Iν cos θdΩ or dE = Iνcos θdAdtdΩdν normal d� � dA Integrate over all angles to obtain net (spectral) flux Z Z π Z 2π Fν = Iν cos θdΩ = Iν(θ; φ) cos θ sin θdθdφ 4πster θ=0 0 Note: 1. For isotropic radiation Iν (independent of angle) the net flux is zero; there is as much energy crossing dA in the ~n direction as in the −~n direction. R −1 −2 2. Total integrated flux F := Fνdν [erg s cm ]. 7 6 Constancy of Specific Intensity along Line of Sight Consider any ray and two points along it: Energy carried through dA1 and dA2: dE1 = Iν1dA1dtdΩ1dν1 dE2 = Iν2dA2dtdΩ2dν2 with dν1 = dν2, and dΩ1 solid angle subtended by dA2 at dA1, and vice versa: 2 dΩ1 = dA2=R 2 dΩ2 = dA1=R dA dA 1 2 R 8 Energy conservation implies dE1 = dE2, i.e. dA dA I dA dt 2 dν = I dA dt 1 dν ν1 1 R2 ν2 2 R2 ) Iν1 = Iν2=const, or dIν=ds = 0 where ds length element along ray. Implication: Spectral intensity is the same at the source and at the detector. Can think of Iν in terms of energy flowing out of source or as energy flowing into detector. 9 7 Constancy of Specific Intensity and Inverse Square Law? Intensity = const. along ray does not contradict inverse square law! Consider sphere of uniform brightness B. Flux measured at point P = flux from all visible points of sphere: Z Z 2π Z θc F = I cos θdΩ = B dφ sin θ cos θdθ 0 0 with dΩ = sin θdθdφ (spherical coordinates), and sin θc = R=r. Thus � P R �c r Z arcsin(R=r) 1 R2 R2 1 F = 2πB sin θ cos θdθ = 2πB = πB / 2 0 2 r r r R α 1 2 using 0 sin x cos xdx = 2 sin α. ) Inverse-Square Law is consequence of decreasing solid angle of object! 10 8 Energy Density uν and Mean Intensity Jν Radiative (Specific) Energy Density uν := energy per unit volume per unit fre- quency range. Radiative energy density uν(Ω) per unit solid angle is defined via dE =: uν(Ω)dV dΩdν For light, volume element dV = cdt · dA, so dE = cuν(Ω)dAdtdΩdν Comparison with definition of intensity dE =: IνdAdtdΩdν implies uν(Ω) = Iν=c 11 The specific energy density uν then is Z 1 Z 4π u = u (Ω)dΩ = I (Ω)dΩ =: J ν ν c ν c ν where the mean intensity is defined by 1 Z J := I (Ω)dΩ ν 4π ν Note that for an isotropic radiation field, Iν(Ω) = Iν, Iν = Jν. Total radiation energy density [erg cm−3] by integration Z 4π Z u = u dν = J dν ν c ν 12 9 Photons and Intensity • Total number of photons (states) in phase space: Z g N := nV = f d3p d3x h3 with n = N=V photon density, f=distribution function aka phase space oc- cupation number, g=degeneracy factor measuring internal degree of freedom. (Heisenberg uncertainty principle ∆x ∆p ∼ h; one particle mode occupies a volume h3; g = 2 for photon with spin ±1 [left-/right-handed circularly polarized]). In phase space element d3p dN = 2fd3x h3 • Energy flow through volume element dV = d3x = dA (cdt): d3p dE = hνdN = 2fhνdA(cdt) h3 hν • For photons jpj = c , and using spherical coordinates h3 d3p = p2dpdΩ = ν2dνdΩ c 13 so that 2hν3 dE = fdAdtdΩdν c2 • Compare with definition of specific intensity dE =: IνdAdtdΩdν then 2hν3 I = f ν c2 3 3 • Note: Since Iν/ν / f, Iν/ν is Lorentz invariant because phase space distribution f / dN=d3xd3p is invariant (since phase space volume element 3 3 dVps := d xd p is itself invariant). 14 10 Radiation Transfer • Transport of radiation ("radiation transfer") through matter is affected by emission and absorption. Specific intensity Iν will in general not remain constant. E2 E E E E E E 1 A B C Figure 1: Interaction of photons with two energy states of matter (E1 and E2, E2 > E1). (A) stimulated ab- sorption, (B) spontaneous emission, and (C) stimulated emission. Often "absorption" is taken to include both, true absorption (A) and stimulated emission (C), as both are proportional to the intensity of the incoming beam. The "net absorption" could thus be positive of negative, depending on which process dominates. 15 • Emission: Radiation can be emitted, adding energy to beam dE =: jνdVd Ωdtdν −3 −1 −1 −1 where jν [erg cm s ster Hz ] is (specific) coefficient for spontaneous emission = energy added per unit volume, unit solid angle, unit time and unit frequency. Note that jν depends on direction. Comparison with specific intensity dE =: IνdAdtdΩdν gives changes in in- tensity, using dV = dAds, ds distance along ray dIν = jνds If emission is isotropic 1 j =: P ν 4π ν with Pν radiated power per unit volume and frequency. 16 • Absorption: Radiation can also be absorbed, taking energy away. Consider medium with particle number density n [cm−3], each having effec- 2 tive absorbing area= cross-section σν [cm ]: { Number of absorbers: ndAds { Total absorbing area: nσνdAds ) Energy absorbed out of beam: −dIνdAdtdΩdν = IνdAabsorbdΩdtdν = Iν(nσνdAds)dΩdtdν Thus change in intensity: dIν = −nσνIνds =: −ανIνds −1 where αν = nσν is absorption coefficient [cm ]. 17 • Equation of Radiation Transfer: Combing emission and absorption dI ν = j − α I ds ν ν ν • Example I: Pure emission, αν = 0, so dI ν = j ds ν Separation of variables and assuming path starts at s = 0 gives Z s 0 0 Iν(s) = Iν(0) + jν(s )ds 0 Brightness increase is equal to integrated emission along line of sight. • Example II: Pure absorption, jν = 0, so dIν = −ανIνds Assuming path starts at s = 0 gives Z s 0 0 Iν(s) = Iν(0) exp − αν(s )ds 0 Brightness decreases exponentially with absorption along line of sight. 18 • Optical Depth: The optical depth τν is defined by Z s Z s 0 0 0 0 τν(s) := αν(s )ds = n(s )σνds = nσνs 0 0 where last step is valid for a homogeneous medium. With this definition, for pure absorption (jν = 0): −τν Iν(s) = Iν(0)e Medium is said to be optically thick or opaque when τν > 1, and optically thin or transparent when τν < 1. 19 • Mean Optical Depth and Mean Free Path of Radiation: "Mean free path" as equivalent concept; have exponential absorption law: ) Probability of photon to travel at least an optical depth τν: exp(−τν) ) Mean optical depth traveled Z 1 −τν 1 < τν >:= τν exp(−τν)dτν = −e (τν + 1) 0 = 1 0 Average distance a photon can travel without being absorbed = mean free path, < `ν >, determined by ! < τν >= nσν < `ν >= 1 Thus 1 < `ν >= nσν 10 33 • Example: Sun (Rs = 7 × 10 cm, M = 2 × 10 g), average den- 3 sity ρ ∼ M =V ' 1:4 g/cm . Assume Sun is made up predominately of ionized hydrogen, mean number density of electrons (same as for protons) 24 3 n ∼ ρ/mp ∼ 10 particles/cm . A photon in the sun scattering on free −25 2 electrons (Thomson cross-section σT = 6:65 × 10 cm ), thus travels on average only a distance < ` >∼ 1/nσT ∼ 1 cm before being scattered... 20 11 Formal Solution to Radiation Transfer Equation Have dI ν = j − α I ds ν ν ν R s 0 0 With τν(s) := 0 αν(s )ds viz.